Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures that show the spectrum of prime distribution indeed "contains" the information of zeta zeros.
But I had not seen a rigorous proof of this. I am looking for such a proof. Be specific, I am looking for a proof which shows that the peaks of the spectrum of prime distribution are the locations of zeta zeros.
Maybe such a proof exist, can anyone share a link or give some reference on such a proof ?